It all started with the study of the Hilbert transform in terms of scattering...

In the late seventies, Cotlar and I began a systematic study of algebraic scattering systems, and the invariant forms acting on them.

In the late eighties we started working in multidimensional scattering-although many did not consider such approach as relevant.

In the late nineties our outlook was finally vindicated. Multidimensional abstract scattering systems appeared as couterparts of the input-output conservative linear systems.

2. The Hilbert transform

The Hilbert transform operator

is given by convolution with the singular kernel

for . (Analogues in and )

The basic result of Marcel Riesz (1927) is

is bounded on .

Similar boundedness properties are valid in the "weighted" cases, both for and for the iterated ,

where are general measures.

3. The scattering property of the analytic projector

Given we can decompose as , where is analytic and is antianalytic.

Under this decomposition the Hilbert transform can be written as

The analytic projector, associated with the Hilbert tranform operator , is defined as

The crucial observation is that supports the shift operator.

Then, the range of is the set of analytic functions, and its kernel is the set of antianalytic functions:

The "scattering property" of the -dimensional Hilbert transform provides the framework for a theory of invariant forms in scattering systems, leading to two-weight -boundedness results for .

The scattering properties are also essential to providing the two-weight - boundedness of the product Hilbert transform in product spaces, where the analytic projectors supporting the -dimensional shifts are at the basis of the lifting theorems in abstract scattering structures.

Notice that this fact, valid for the product Hilbert transforms, is not valid for the -dimensional Calderón-Zygmund singular integrals, which do not share the scattering property.

4. is bounded on

In fact, is an isometry on , since

and this follows easily from the Plancherel Theorem for the Fourier transform.

The result can also be obtained through the Cotlar Lemma on Almost Orthogonality, which extends the Hilbert transforms into ergodic systems.

These are two different ways to deal with the boundedness of in . The same happens for .

5. Two ways for to

Start checking that is weakly bounded in , and apply the Marcinkiewicz Interpolation Theorem between and , and then, by duality, pass from to .

The "Magic Identity" for given by

is valid for all functions smooth and with compact support.

Using extrapolation, since for implies , then implies , and implies .

The boundedness of in , follows by duality, and interpolation gives the boundedness of in

By polarization, the Magic Identity for an operator becomes

The Magic Identity, and similar ones have been used extensively in harmonic analysis, in particular by Coifman and Meyer. Cotlar and Sadosky, and Rubio de Francia, used the Identity in dealing with the weighted Hilbert transform in Banach lattices.

Gian-Carlo Rota used three different "magic indentities" in his work in combinatorics, and his school encompassed all particular cases in a general inequality.

Gohberg and Krein showed that the polarized Magic Inequality holds in the space , and deduced the theorem of Krein and Macaev in a way similar to the passage from to described before.

The "magic indentities" hold in a variety of non-commutative situations, starting with the non-commutative Hilbert transforms in von Neumann algebras, and that theory has been developed in the last years.

6. is bounded in

6.1. Helson-Szegtheorem (1960).

where are real-valued bounded functions, such that

is equivalent to

(with a special norm).

6.2. Hunt-Muckenhoupt-Wheeden theorem (1973).

is equivalent to

6.3. Remark.

Take note that, although both conditions are necessary and sufficient, the first one is good to such weights, while the second one is good at them.

7. Definition of -boundedness

An operator acting in a Banach lattice is u-bounded if

The -boundedness of operators is considerably weaker than boundedness. For example,

is -bounded on if and only if

.

Now we can translate another equivalence for the Helson-Szeg theorem for p=2: